Problem 70 - Algebra and Trigonometry

Question

Solve the inequality. \(x^{2} e^{x}-2 e^{x}<0\)

Step-by-Step Solution

Answer

The solution is \(-\sqrt{2} < x < \sqrt{2}\).

Step 1: Factor the Expression

The inequality given is \(x^{2} e^{x}-2 e^{x}<0\). Notice that \(e^{x}\) is a common factor. Factor it out to rewrite the inequality as: \(e^{x}(x^{2} - 2) < 0\).

Step 2: Analyze the Exponential Term

The term \(e^{x}\) is always positive for all real values of \(x\), as the exponential function is never zero or negative.

Step 3: Solve the Quadratic Expression

Since \(e^{x}\) is positive, the inequality sign depends on \(x^{2} - 2 < 0\). This reduces to \(x^{2} < 2\).

Step 4: Solve the Quadratic Inequality

For \(x^{2} < 2\), taking the square root of both sides, we obtain \(-\sqrt{2} < x < \sqrt{2}\).

Step 5: Combine the Results

Since \(x\) is restricted to \(-\sqrt{2} < x < \sqrt{2}\) for the inequality to be satisfied, the solution to the inequality is this interval.

Key Concepts

quadratic inequality

Quadratic inequalities involve expressions like \(x^2\) and are solved by finding the values of \(x\) that make the expression true. For example, take \(x^2 < 2\). We're looking for \(x\) values where squaring \(x\) gives a result less than 2. This inequality is like asking, "Which numbers, when squared, are under the limit of 2?"

To solve this, we find the boundary points by setting \(x^2 = 2\), leading to \(x = \pm\sqrt{2}\). This tells us where \(x\) reaches the value of 2. The inequality sign (<) indicates that we need values strictly less than 2. So, our solution is an interval between these points.

Thus, the solution to \(x^2 < 2\) is the open interval \(-\sqrt{2} < x < \sqrt{2}\). This means any number between -\(\sqrt{2}\) and \(\sqrt{2}\), but not including them, will satisfy the inequality.

exponential functions

Exponential functions, like \(e^x\), have unique properties. The function \(e^x\) is always positive, regardless of \(x\). This is because the exponential function grows rapidly as \(x\) becomes positive and decreases towards zero as \(x\) becomes negative. However, it never actually reaches zero.

In solving inequalities where \(e^x\) is a factor, you can assume it doesn't change the inequality direction. For \(e^x(x^2 - 2) < 0\), the critical factor is \(x^2 - 2\) because \(e^x > 0\). Knowing this helps to correctly interpret the inequality by focusing on the behavior of the quadratic expression without worrying about the exponential part.

Understanding this property of exponential functions simplifies many types of mathematical problems, as you know one part of your expression is consistently positive.

factoring expressions

Factoring is the process of breaking down expressions into simpler "multiplied" components. Consider the expression from our problem: \(x^2 e^x - 2 e^x\). Here, \(e^x\) is a common factor, which can be factored out to simplify the expression to \(e^x(x^2 - 2) < 0\).

Why factor? Once you factor an expression, it becomes much easier to analyze and solve. In our case, factoring helps isolate the term \(x^2 - 2\), so we can focus on determining when this part results in negative values, given that \(e^x\) won't affect the direction of the inequality.

Factoring is a key technique in algebra that simplifies complex problems. It breaks down expressions, making them more manageable, and lays out a clearer path to finding solutions. Practicing factoring makes tackling math challenges more straightforward, especially when dealing with inequalities or polynomials.